WASHINGTON — Supporters of bipartisan immigration legislation smoothed the way Friday for likely Senate passage of their handiwork, overcoming last-minute disagreements at the bill’s controversial core and tacking on other items certain to build support.

A test vote was set for Monday on the bill, which calls for a military-style surge to increase security at the U.S-Mexican border. At the same time it sets out a 13-year pathway to citizenship for millions of immigrants living in the United States unlawfully.

Sen. Lisa Murkowski of Alaska became the 11th Republican to announce her support for the legislation in the Democratic-controlled Senate. More were expected to follow, possibly enough to produce 70 votes or more and easily overwhelm its critics.

Some Democrats said a heavy show of support at the end of next week could alter the bill’s trajectory in the House, where majority Republicans strongly oppose citizenship for immigrants who came to the country illegally or overstayed their visa.

“Hopefully as congressmen look how their senators voted, they will be influenced by it,” said Sen. Chuck Schumer, D-N.Y., who has played a major role for Democrats on the issue.

The bill’s critics made no claim they could block it in the Senate, but said their position would be vindicated in the long run.

Republican Sen. Jeff Sessions of Alabama said the measure’s claims of border security were no different than previous assurances. “Time and again, politicians have promised, promised, promised. But they never delivered, delivered, delivered. And that’s a fact,” he said.

With immigration at the top of President Barack Obama’s second-term domestic agenda, White House spokesman Jay Carney labeled the Senate agreement a breakthrough. He refrained from issuing an outright endorsement of the legislation, even though Cabinet secretaries were consulted on some portions of it and administration officials drafted others.

The day’s developments marked a victory for the Senate’s so-called Gang of Eight, four Democrats and four Republicans who spent months working out the basic framework of immigration legislation. They then warded off unwanted changes in the Senate Judiciary Committee last month, and in recent days, negotiated significant alterations with a group of Republicans who were uncommitted but willing to swing behind the bill if it were changed.

The principal demand was for tougher border security, particularly after the Congressional Budget Office estimated this week the bill would fail to prevent a future buildup in the population of immigrants in the country illegally.

Republican Sens. John Hoeven of North Dakota and Bob Corker of Tennessee, who had spent about a week negotiating with members of the Gang of Eight for changes, announced the agreement on Thursday. A day later, Corker said in the Senate the bill is a chance to deal with “the issues of security many of our citizens across the country care about, but at the same time allow 11 million people to come out of the shadows and work in the light and be a part of this great, great nation.”

The result of the negotiations was a series of expensive and highly detailed steps to guard against future illegal immigration across the 2,000-mile border with Mexico.

There are similar specifications for points of entry from Mexico. At the one in San Diego, the government is mandated to install two nonintrusive inspection systems; one radiation monitor and one detection and classification network.

The legislation also requires a doubling of the Border Patrol, with the hiring of 20,000 new agents, the purchase of 12 new unmanned surveillance drones and the construction of 350 miles of new fencing, to bring the total to 700 miles.

Other provisions in the bill would require employers to verify the legal status of their workers, before they are hired and then periodically afterwards. A biometric system would be phased in at 30 airport crossings to track the comings and goings of foreigners.

Most controversially, the measure creates a chance at citizenship for immigrants who are in the country illegally. It also sets up a new temporary program for farm workers, and new visa programs for workers recruited for the technology industry and those of lesser skills.

The new security provisions would be put in place over a decade, in line with the 10-year path to a permanent resident green card that the bill sets out for immigrants in the U.S. illegally. During that time, the immigrants could live and work legally in a provisional status.

As part of late negotiations, the bill makes clear that no immigrant can get credit for payroll taxes paid when they lacked legal status. Credits are used to determine the level of Social Security benefits workers are entitled to in retirement.

Sen. Bernie Sanders, a Vermont independent, joined by several Democrats, secured a $1.5 billion temporary jobs program for low-income youth. The funds will come from a temporary $10 surcharge imposed on visa applications from companies hiring guest workers and international workers who receive green cards.

Two provisions designed to aid Alaska seafood processers were backed by the state’s senators, Democrat Mark Begich as well as Murkowski. The first permits the companies to hire foreign students visiting the U.S. on so-called J-1 visas. The effect is to overturn a recent ruling by the departments of State and Labor that banned the practice. The second declares Alaska fish processing as a “shortage occupation,” which would expedite the industry’s ability to recruit seasonal workers outside the U.S. through a new W visa program set up in the bill.